Kostka polynomial

In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory.

The two-variable Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials. Here the indices λ and μ are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial Kλμ(t) = Kλμ(0, t).

There are two slightly different versions of them, one called transformed Kostka polynomials.

The one variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials Pμ to Schur polynomials sλ:

s_\lambda(x_1,\ldots,x_n) =\sum_\mu K_{\lambda\mu}(t)P_\mu(x_1,\ldots,x_n;t).\

The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by Pμ) to Schur polynomials sλ:

s_\lambda(x_1,\ldots,x_n) =\sum_\mu K_{\lambda\mu}(q,t)J_\mu(x_1,\ldots,x_n;q,t)\

where

J_\mu(x_1,\ldots,x_n;q,t) = P_\mu(x_1,\ldots,x_n;q,t)\prod_{s\in\mu}(1-q^{a(s)}t^{l(s)}).\

Kostka numbers are special values of the 1 or 2 variable Kostka polynomials:

K_{\lambda\mu}= K_{\lambda\mu}(1)=K_{\lambda\mu}(0,1).\

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